1201.5411 (Marco Dalai)

Sphere Packing and Zero-Rate Bounds to the Reliability of
Classical-Quantum Channels    [PDF]

Marco Dalai

In this paper, the sphere packing bound of Fano, Shannon, Gallager and Berlekamp and the zero-rate bound of Berlekamp are extended to general classical-quantum channels. The sphere packing upper bound for the reliability function, for the case of pure-state channels, coincides at high rates with a lower bound derived by Burnashev and Holevo \cite{burnashev-holevo-1998}. Thus, for pure state channels, the reliability function at high rates is now exactly determined. For the general case, the obtained upper bound expression at high rates was conjectured to represent also a lower bound to the reliability function, but a complete proof has not been obtained yet. The exact expression of the zero-rate reliability for a general classical-quantum channel with no zero-error capacity is obtained as a straightforward extension of Berlekamp's theorem for classical channels. A bound for the low rate region obtained by Blahut is considered for the case of pure and non-pure state channels, also reviewing the applicability of the bound in the classical case.

View original: http://arxiv.org/abs/1201.5411